Effects of numerals vs. variables on students’ expression transformation processes and strategies

• In Event: Poster Session 12
  In Poster Session: PS 12 Topic - Education, Schooling

Abstract

Although numerals and variables often follow the same principles of arithmetic operations (e.g., 3+3=2⋅3 and x+x=2x), numerals represent specific values that can be computed to a number (e.g., 2⋅3=6) whereas variables represent unknowns that can not be simplified to a value. Prior work revealed students’ struggles with variables (Küchemann, 1980), and that the notion of unknown interferes with algebra problem solving (Malisani & Spagnolo, 2008). Here, we examined whether middle-school students’ expression transformation processes and strategies varied between expressions presented in numerals versus variables (pre-registered with OSF).
We utilized a dataset in which middle-school students completed a two-hour intervention using a web-based dynamic algebra notation system to transform expressions into a mathematically equivalent goal (Figure 1a). Students had unlimited opportunities to reset and reattempt problems, and the system captured and time-stamped all student actions in the expression transformation process (Figure 1b). The analytic sample included 125 students who completed four expression transformation problems—two pairs that aligned on the structure but varied in whether numerals or variables were used in the expressions (Table 1). We compared students’ attempts and steps (i.e., valid transformations) on two pairs of variable vs. numerical problems to explore how symbols influence their problem-solving processes and strategies. Because the minimum number of steps to reach the goal varied across problems, we calculated students’ step efficiency (i.e., the minimum steps / students’ solution steps) on each problem to compare the efficiency of their final solutions.
Preliminary analysis revealed that, compared to numerical problems, students were less likely to complete variable problems on their first attempt, reattempted variable problems more frequently, and took more steps on variable problems. However, students’ final solutions were more efficient (i.e., involved fewer steps) on variable, as opposed to numerical, problems. We conducted a repeated measures MANOVA on the four measures to compare variable and numerical problems in Pair 1, and found that the differences between these two problems were significant on all four measures, ps<.001, 𝜂p2s>.248. We repeated the MANOVA on Problem Pair 2 and replicated the findings for all except step efficiency (p=.063), ps<.001, 𝜂p2s>.089.
Consistent with the documented challenges with variables, students struggled with transforming expressions presented in variables. However, they implemented more efficient final solutions on variable, compared to numerical, problems. Although numbers and variables follow similar arithmetic principles, variables limit some arithmetic operations (e.g., 3+2=5 vs. x+y =x+y) and may minimize students’ impulse for immediate computations that lead to inefficient solutions. When computations are constrained in variable problems, students may be more likely to notice the problem structure and find efficient solutions. The study leverages fine-grained data from learning technologies to examine students’ thinking of mathematical symbols, and the findings provide insights into designing instructional activities. By replacing numerals with variables in algebraic expressions, educators may be able to guide students’ attention to mathematical structures, support efficient solutions, and minimize applying computations or standard procedures by rote. Data collection will continue in fall 2020, and results on additional pair problems and in-application measures will be discussed.

Authors

• Jenny Yun-Chen Chan, TheEducation University of Hong Kong
  Presenting Author

• Erin Ottmar, Worcester Polytechnic Institute
  Non-Presenting Author

• Hannah Smith, Worcester Polytechnic Institute
  Non-Presenting Author

• Avery Harrison Closser, Worcester Polytechnic Institute
  Non-Presenting Author

• Kathryn Christine Drzewiecki, Worcester Polytechnic Institute
  Non-Presenting Author